A Statistical Factor Asset Pricing Model Versus the 4-Factor Model/Modelo Estatistico de Aprecamento de Ativos Versus o Modelo de Quatro Fatores.
| Autor | Santana, Veronica de Fatima |
| Cargo | Texto en ingles - Ensayo |
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Introduction
This research aims to analyze how well a statistical factor model prices Brazilian stock returns compared to the 4-factor model as developed by Fama and French (1993, 1996), and Carhart (1997). To do so, we employ a Principal Component Analysis in a set of stocks listed on the Sao Paulo Stock Exchange (B3), extracting latent common components which are then considered as risk factors in a pricing model. We test the models' performance for pricing industry portfolios, analysing the implications of they being true in both the time series dimension, that is, investigating whether all intercepts are zero --risk factors can explain assets' returns over time--and in the cross-section dimension, analyzing whether the cross-section intercepts are zero--risk factors can explain the differences in returns among different assets--and whether the risk premia are positive according to the Fama-Macbeth Fama and MacBeth (1973) procedure.
For several decades, the theoretical and empirical research in Finance has been seeking to understand what drives investment decisions and, consequently, to derive the laws that govern market prices. Starting with the mean-variance analysis of Markowitz (1952) and the development of the CAPM by Sharpe (1964), Lintner (1969) and Black (1972), the asset pricing literature comes from a long way. While in the decades of 1960 and 1970 the CAPM was empirically validated, the next decades brought the documentation of several anomalies, that is, several patters of expected returns the CAPM framework was not able to explain. These anomalies gave rise to a line of research on multi-factor asset pricing models, which are mostly empirically motivated; notwithstanding, they can be fitted into the more sophisticated theoretical framework of the Intertemporal CAPM Merton (1973) and the Arbitrage Pricing Theory (APT) Ross (1976), that allows for multidimensional sources of risk.
The CAPM is an attractive model to explain how market risk is related with expected returns and investment risk, however early empirical analyses provided rather poor estimates. In order to advance in the subject, Fama and French (1993, 1996) propose a model with three factors to explain market returns: the original CAPM market return in addition to book-to-market ratio (BTM) and size. The authors explain that these two additional factors are able to capture the major variances that are not explained solely by the market return. Later, Jegadeesh and Titman (1993) observe that past returns of market assets also bear significant trends that are able to explain further returns, i.e., they find that shares with higher past returns are more likely to provide abnormal positive returns, in comparison with shares with lower past returns. From the findings on this additional factor, so called momentum, Carhart (1997) develops a deeper analysis of this effect on empirical predictions, so to propose its inclusion as a fourth factor on the Fama and French (1993, 1996) 3-factor model, yielding the well-known 4-factor asset pricing model. It has been widely applied in several studies, especially on the investigation of additional variables affecting asset prices, and on the assessment of most powerful versions of the model.
While the empirically chosen factors is the dominant approach to identify relevant risk components, other researchers have proposed different approaches, such as chosing macroeconomics variables as risk factors (Chen et al. (1986)) and even statistically-chosen factors. Arguing on the advantage of statistical techiniques to find underlying, or latent, variables correlated with the movement of returns, some suggest applying Principal Component Analysis (PCA) for identifying common risk factors, such as Roll and Ross (1980) and Connor and Korajczyk (1988). In a PCA, existing n variables are reduced through orthogonal transformation into a set of m [less than or equal to] n linear uncorrelated factors bearing a relevant share of the variance of the original variables.
Studies show that this approach improves the estimation of large covariance matrices (Alexander (2000)), and provides portfolios with lower risk, in comparison with empirically-identified factors (Engle et al. (1990)).
Despite the simplicity advantage of the use of PCA for portfolio composition, its application on the analysis of asset pricing in Brazil is lacking, for the predominant approach is still the 4-factor model. Studies show that the variances on asset returns are explained by the 4 factors on the baseline model, however the significance of each factor is varying across studies. Rogers and Securato (2009) analyze the Brazilian market and compare three models: the original CAPM, Fama-French modification with 3 factors, and the Reward Beta model from Bornholt (2007). They find that the Fama-French version of CAPM carries higher explanatory power to drive assets returns, however the BTM factor is not significant. Matos and Rocha (2009) analyze the fit of the baseline 4-factor pricing model on the returns of mutual funds in the Brazilian market. Their results indicate that the performance of Brazilian mutual funds regarding pricing and returns forecasts depends not only on the model factors, but also on particular features of each fund. In special, Matos and Rocha (2009) find that the 4-factor model provides significant estimates for investment funds with higher stockholder equity and larger deviation from the market risk parameter, while it does not provide appropriate estimates for other types of mutual funds.
Chague (2007) analyzes the application of the original CAPM and Fama-French modification with 3 factors in Brazilian stock market for the period of 1999-2007, and compares the outcomes with U.S. stock market for the same period, as a benchmark comparison. The results indicate the Fama-French approach has stronger explanatory power in Brazil, however it provides loose estimates in comparison with results in U.S. market. Under the analysis, this outcome is attributed to the absence of clear-cut anomalies in Brazilian firms regarding firms size. Mussa et al. (2012) analyze the application of the 4-factor model for the returns on Brazilian stock market from 1995 to 2006, considering twelve portfolios. Results indicate the traditional market factor is always significant, however it does not explain comprehensibly the variation in stock returns. In this line, they explain the four factors have complementary explanatory power in the Brazilian context.
When comparing a statistical factor model and the 4-factor model, we found, in general, that the latter is better able to prices assets in the time-series dimension. The GRS statistic (Gibbons et al. (1989), Campbell et al. (1997)) for the 4-factor model is smaller that the statistical factor model's (1.062 versus 2.794), so that it does not reject the null hypothesis of jointly zero time-series intercepts, while the statistical factor model does. Regarding the ability to explain why different assets have different levels of returns, both models generate alphas statistically indistinguishable from zero, indicating that no returns are left unexplained; however, none of the risk premia estimates are positive for both models. Although, again, the 4-factor model seems closer to attend the cross-section implications than the statistical model, all t statistics are outside the rejection area. However, it is necessary to point the presence of survivorship bias in the data as a limitation of this research. For the PCA analysis we need continuous series so we could only include active stocks in the analysis, so we can only conclude about the models' abilities to price surviving assets.
This paper provides two main contributions to the literature of asset price modeling in Brazil. First, it refers to a distinct analysis when comparing to the predominant approach of the 4-factor model. We show that returns can also be estimated by means of statistical factors, rather than empirically-supported factors. Although this approach deviates from the traditional CAPM-empirical rationale, it is consistent with the abstract theoretical view of the factor model construction, for it shows that the movement of variables is explained by a few factors bearing a relevant portion of variance. Second, we complement the literature on asset pricing in Brazil, for we provide new results that are comparative to the investigation of which empirical factors are relevant to explain investment returns in the Brazilian market.
The remaining of the paper is structured as follows. Section 2 reviews the asset pricing literature from the CAPM, passing through the time-series and cross-section tests of the model, to the multifactor models. Section 3 describes the data and the multifactor tests, while section 4 compares the time-series and cross-section performance of the models. Finally, section 5 gives some concluding remarks.
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Factor Pricing Models
Suppose i = 1, 2, ..., I investors and j = 1, 2, ..., N assets, the decision to maximize the terminal wealth [[??].sub.i,1] investing both in risky assets and in a risk-free asset [R.sub.f], given the initial wealth [w.sub.i,0] as the budget restriction, the investor's first order condition is
Cov[[u'.sub.i]([[??].sub.i,1]), [[??].sub.j]] = -E[[u'.sub.i]([[??].sub.i,1])][E([[??].sub.j]) - [R.sub.f]], (1)
Assuming either joint normality or quadratic utility, equation (1) yields the Capital Asset Pricing Model (CAPM) with the aggregate wealth portfolio as the single risk component for any risky asset j:
[mathematical expression not reproducible] (2)
which is credited to Sharpe (1964) and Lintner (1969).
In the following decades, the CAPM went through a battery of tests to evaluate its empirical validity. Despite the critique of Roll (1977) that the CAPM's theory cannot be empirically tested, several authors developed strategies and statistics to test the relationship between...
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